Descartes' "Kissing Circle" Theorem relates the radii, $r_1$, $r_2$, $r_3$, $r_4$, of four mutually-tangent circles thusly: $$( k_1 + k_2 + k_3 + k_4 )^2 = 2 ( k_1^2 + k_2^2 + k_3^2 + k_4^2 ) \qquad\text{where } k_i := \pm\frac{1}{r_i}$$ with the "$\pm$" sign indicating internal or external tangency of the corresponding circle.
Suppose, given three mutually-tangent circles, we determine the external fourth kissing circle, and then we want to find out the radius of a 5th circle that touches the 4th circle and any two of the circles of the original three circles. How do we find out the radius of this 5th circle?